Gordon Strachan's side have failed to win any of their previous 15 away games in the competition and their travel sickness continued after losing out to Marcos Senna's second-half free-kick at El Madrigal.
The result means the Hoops must now realistically take something from their forthcoming double-header against European champions Manchester United to qualify from Group E.
Celtic, who were held at home by Aalborg BK in their first game, started brightly with the recalled Shaun Maloney and Aiden McGeady causing panic in the home defence with direct running.
Maloney tested Diego Lopez with a long-range free-kick, while Sebastian Eguren headed a clear chance wide at the other end.
Celtic were then indebted to Artur Boruc for keeping the scores level, as the Poland keeper snuffed out a chance for Giuseppe Rossi when the former Manchester United striker was through one-on-one.
Boruc repelled a header from Joseba Llorente and then tipped over Rossi's powerful shot as the home side - who finished second behind Real Madrid in La Liga last season - began to find their attacking feet.
However, it was the visitors who spurned the best chance of the first half as Georgios Samaras saw his shot somehow clawed away by Lopez, after Diego Godin's errors had allowed the Greek striker in on goal.
Villarreal started the second half intent on forcing a breakthrough and the visitors were pinned back and forced to rely on swift breaks to ease the pressure.
However, the Celtic defence was finally breached on 67 minutes when Paul Hartley fouled Llorente 25 yards from goal and Spain midfielder Senna curled the ball over the wall and past the diving Boruc.
Substitute Scott McDonald came close to restoring parity with a late dipping volley which found the roof of the net.
But despite plenty of effort from the Hoops, they failed to find the breakthrough and are now facing an uphill battle to qualify from a group which is now bossed by two sides expected to go a long way in this year's competition.